Finally, we need to know the explicit form of \(\eth \)  in order to compute the equivariant index. Its form depends on the choice of \(V\)  which is commonly taken to be \(V\propto \langle \Psi ,\overline{\mathcal {Q}\Psi }\rangle \) , where \(\Psi \)  denotes the fermionic field content, \(\langle \cdot ,\cdot \rangle \)  an inner product invariant under the symmetries \(L\)  and the bar denotes complex conjugation in a suitable sense such that \(\langle \mathcal {Q}\Psi ,\overline{\mathcal {Q}\Psi }\rangle \)  is positive semi-definite. For the gauge theory-part (after gauge-fixing) it was shown in [1]} that \(\eth \)  is determinedTo be precise, there are also terms with lower orders in derivatives present in the one-loop piece of \(\mathcal {Q}V\) . However, the index only depends on the symbol of \(\eth \)  to which these terms have no contribution. by the following action on the bosonic fields:
\(\mathrm {d}^\dagger (\iota _vF+\mathrm {d}(\cos \omega \Phi )),\qquad \mathrm {d}^\dagger A,\qquad P^+_\omega (F+\iota _v\star \mathrm {d}\Phi ),\) 
